Advanced Research on Second-Order Variational Analysis with New Applications to Optimization, Control, and Practical Modeling

该项目致力于开发各种应用中出现的大型系统的数学分析和优化的先进工具。这样的系统包括非标准的优化相关和平衡问题，数据在通常意义上可能是不可微的。除了开发新的数学知识，首席研究员和他的七名参与该项目的博士生（包括四名来自数学科学中代表性不足的群体的学生）特别关注社会经济学、交通平衡、行为科学和水资源等应用中出现的各种模型。该项目主要研究二阶变分分析、优化和系统控制方面的新课题，这些课题主要是由应用中出现的问题引起的。它有条件地分为五个相互关联的部分。第一部分关注二阶广义微分理论中的开放问题及其在非线性分析中优化相关领域的应用。特别注意了重要的扩展实值函数类的主要二阶广义导数的构造计算，这在项目的后续部分起着至关重要的作用。第二部分致力于各种稳定性概念的二阶表征，包括非多面体二次规划中的倾斜稳定性和完全稳定性，以及椭圆变分不等式。该项目的第三部分涉及变分系统中关键乘子的研究，这些乘子主要负责优化的原始对偶算法的缓慢收敛。在这里，PI和他的合作者开发了新的牛顿型方法来解决非光滑优化和相关问题。第四部分致力于控制理论的新发展，包括反馈控制和稳定的ODE和PDE系统。考虑的另一个主要主题是在后续应用中起关键作用的扫描过程的非凸版本的最优控制。其中一些应用是该项目第五部分的重点，其中PI和他的合作者对控制平面人群运动模型的优化问题进行了全面的研究，这在社会经济学和交通平衡中得到了很好的认可。考虑的其他类型的应用源于行为科学和水资源模型。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is devoted to developing advanced tools of mathematical analysis and optimization of large-scale systems arising in various applications. Such systems include nonstandard optimization-related and equilibrium problems with data that may not be differentiable in the usual sense. Besides developing new mathematical knowledge, the principal investigator and his seven PhD students who participate in the project (including four students from underrepresented groups in the mathematical sciences) pay particular attention to a variety of models arising in applications from socioeconomics, traffic equilibria, behavioral science, and water resources.The project studies new topics in second-order variational analysis, optimization, and systems control that are largely motivated by problems arising in applications. It is conditionally divided into five interrelated parts. Part I concerns open problems in second-order generalized differential theory and its applications to optimization-related areas of nonlinear analysis. Particular attention is paid to constructive computations of major second-order generalized derivatives for remarkable classes of extended-real-valued functions which play a crucial role in the subsequent parts of the project. Part II is devoted to second-order characterizations of various stability notions, including tilt and full stability in nonpolyhedral conic programming, and elliptic variational inequalities. Part III of the project deals with the study of critical multipliers in variational systems that are largely responsible for the slow convergence of primal-dual algorithms of optimization. Here, the PI and his collaborators develop new Newton-type methods to solve nonsmooth optimization and related problems. Part IV is devoted to novel developments in control theory, including feedback control and stabilization of ODE and PDE systems. Another major topic considered is optimal control of nonconvex versions of the sweeping process that plays a key role in subsequent applications. Some of these applications are the focus of Part V of the project, where the PI and his collaborators undertake a comprehensive study of the optimization problems for the controlled planar crowd motion model, which is well recognized in socioeconomics and traffic equilibria. Other types of applications considered stem from behavioral sciences and water resource models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.